Solve for $x$ and $y$ using elimination. $\begin{align*}2x+7y &= 9 \\ 9x+4y &= -9\end{align*}$
Explanation: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-4$ and the bottom equation by $7$ $\begin{align*}-8x-28y &= -36\\ 63x+28y &= -63\end{align*}$ Add the top and bottom equations. $55x = -99$ Divide both sides by $55$ and reduce as necessary. $x = -\dfrac{9}{5}$ Substitute $-\dfrac{9}{5}$ for $x$ in the top equation. $2( -\dfrac{9}{5})+7y = 9$ $-\dfrac{18}{5}+7y = 9$ $7y = \dfrac{63}{5}$ $y = \dfrac{9}{5}$ The solution is $\enspace x = -\dfrac{9}{5}, \enspace y = \dfrac{9}{5}$.